He addresses this sort of question in the intro:
> So it is that Christianity has taught the Christian to think dauntlessly of everything earthly and worldly, including death. It is almost as though the Christian must be puffed up because of this proud elevation above everything men commonly call misfortune, above that which men commonly call the greatest evil. But then in turn Christianity has discovered an evil which man as such does not know of; this misery is the sickness unto death. What the natural man considers horrible -- when he has in this wise enumerated everything and knows nothing more he can mention, this for the Christian is like a jest. Such is the relation between the natural man and the Christian; it is like the relation between a child and a man: what the child shudders at, the man regards as nothing. The child does not know what the dreadful is; this the man knows, and he shudders at it. The child’s imperfection consists, first of all, in not knowing what the dreadful is; and then again, as an implication of this, in shuddering at that which is not dreadful. And so it is also with the natural man, he is ignorant of what the dreadful truly is, yet he is not thereby exempted from shuddering; no, he shudders at that which is not the dreadful: he does not know the true God, but this is not the whole of it, he worships an idol as God.
> Only the Christian knows what is meant by the sickness unto death. He acquires as a Christian a courage which the natural man does not know -- this courage he acquires by learning fear for the still more dreadful. Such is the way a man always acquires courage; when one fears a greater danger, it is as though the other did not exist. But the dreadful thing the Christian learned to know is "the sickness unto death."
It's not just about not fearing death, it's about not fearing death because you fear something much worse.